In this paper, we extend the Ky Fan inequality to several general integral forms, and obtain the monotonic properties of the function L Ls(a,b) s(α−a,α−b) withα, a, b ∈ (0,+∞)and s∈R

http://jipam.vu.edu.au/

Volume 7, Issue 4, Article 130, 2006

GENERALIZATIONS OF THE KY FAN INEQUALITY

AI-JUN LI, XUE-MIN WANG, AND CHAO-PING CHEN JIAOZUOUNIVERSITY

JIAOZUOCITY, HENANPROVINCE

454000, CHINA

liaijun72@163.com wangxm881@163.com

COLLEGE OFMATHEMATICS ANDINFORMATICS

RESEARCHINSTITUTE OFAPPLIEDMATHEMATICS

HENANPOLYTECHNICUNIVERSITY

JIAOZUOCITY, HENAN454010, CHINA

chenchaoping@hpu.edu.cn

Received 25 November, 2005; accepted 06 October, 2006 Communicated by D. ¸Stefˇanescu

ABSTRACT. In this paper, we extend the Ky Fan inequality to several general integral forms, and obtain the monotonic properties of the function _L ^Ls(a,b)

s(α−a,α−b) withα, a, b ∈ (0,+∞)and s∈R.

Key words and phrases: Generalized logarithmic mean, Monotonicity, Ky Fan inequality.

2000 Mathematics Subject Classification. 26A48, 26D20.

1. INTRODUCTION

The following inequality proposed by Ky Fan was recorded in [1, p. 5] : If0 < x_i ≤ ¹₂ for i= 1,2, . . . , n, then

(1.1)

Qn i=1x_i Qn

i=1(1−x_i) _n¹

≤

Pn i=1x_i Pn

i=1(1−x_i), unlessx₁ =x₂ =· · ·=x_n.

With the notation

(1.2) Mr(x) =







1 n

Pn i=1x^r_i¹_r

, r6= 0;

(Qn

i=1x_i)ⁿ¹ , r= 0,

ISSN (electronic): 1443-5756 c

2006 Victoria University. All rights reserved.

The authors were supported in part by the Science Foundation of the Project for Fostering Innovation Talents at Universities of Henan Province, China.

346-05

whereM_r(x)denotes ther-order power mean ofx_i >0fori= 1,2, . . . , n, the inequality (1.1) can be written as

(1.3) M0(x)

M₀(1−x) ≤ M1(x) M₁(1−x).

In 1996, Zh. Wang, J. Chen and X. Li [12] found the necessary and sufficient condition for

(1.4) M_r(x)

M_r(1−x) ≤ M_s(x) M_s(1−x)

whenr < s. Recently, Ch.-P. Chen proved that the function _L ^Lr(a,b)

r(1−a,1−b) is strictly increasing for 0 < a < b ≤ ¹₂ and strictly decreasing for ¹₂ ≤ a < b < 1, wherer ∈ (−∞,∞)andL_r(a, b) is the generalized logarithmic mean of two positive numbersa, b, which is a special case of the extended meansE(r, s;x, y)defined by Stolarsky [10] in 1975. For more information about the extended means please refer to [4, 6, 8, 11] and references therein.

Moreover, we have,

L_r(a, b) = a, a=b;

L_r(a, b) =

b^r+1−a^r+1 (r+ 1)(b−a)

¹_r

, a6=b, r6=−1,0;

L−1(a, b) = b−a

lnb−lna =L(a, b);

L₀(a, b) = 1 e

b^b a^a

_b−a¹

=I(a, b),

where L(a, b) and I(a, b) are respectively the logarithmic mean and the exponential mean of two positive numbersa andb. Whena 6= b, L_r(a, b) is a strictly increasing function ofr. In particular,

r→−∞lim L_r(a, b) = min{a, b}, lim

r→+∞L_r(a, b) = max{a, b}, L₁(a, b) =A(a, b), L₋₂(a, b) =G(a, b),

whereA(a, b)andG(a, b)are the arithmetic and the geometric means, respectively. Fora6=b, the following well known inequality holds:

(1.5) G(a, b)< L(a, b)< I(a, b)< A(a, b).

In this paper, motivated by inequality (1.4), we will extend the inequality (1.4) to general integral forms. Some monotonic properties of several related functions will be obtained.

Theorem 1.1. Let

f_α(s) =

Rb a x^sdx Rb

a(α−x)^sdx

!¹_s

= L_s(a, b) L_s(α−a, α−b),

s ∈ (−∞,+∞)and α be a positive number. Thenf_α(s)is a strictly increasing function for [a, b]⊆(0,^α₂], and is a strictly decreasing function for[a, b]⊆[^α₂, α).

Corollary 1.2. If[a, b]⊆(0,^α₂]andαis a positive number, then a

α−b < G(a, b)

G(α−a, α−b) < L(a, b) L(α−a, α−b)

< I(a, b)

I(α−a, α−b) < A(a, b)

A(α−a, α−b) < b α−a. (1.6)

If[a, b]⊆[^α₂, α), the inequalities (1.6) is reversed.

Corollary 1.3. Lethα(s) = _Rb

ax^sdx Rα−a

α−b x^sdx

¹_s

,s∈(−∞,+∞)andαbe a positive number. Then hα(s) is a strictly increasing function for[a, b] ⊆ (0,^α₂], or a strictly decreasing function for [a, b]⊆[^α₂, α).

In [13], Feng Qi has proved that the function r 7→

1 b−a

Rb ax^rdx

1 b+δ−a

Rb+δ a x^rdx

!¹_r

= L_r(a, b) L_r(a, b+δ)

is strictly decreasing withr∈(−∞,+∞). Now, we will extend the conclusion in the following theorem.

Theorem 1.4. Let

f(s) =

1 b−a

Rb a x^sdx

1 d−c

Rd c x^sdx

!¹_s

= Ls(a, b) L_s(c, d),

s ∈ (−∞,+∞) anda, b, c, dbe positive numbers. Then f(s)is a strictly increasing function forad < bc, or a strictly decreasing function forad > bc.

Corollary 1.5. Let

h(s) =

1 b−a

Rb a x^sdx

1 d−a

Rd a x^sdx

!¹_s

= L_s(a, b) L_s(a, d),

s∈(−∞,+∞)anda, b, dare positive numbers. Thenh(s)is a strictly increasing function for d < b, or a strictly decreasing function ford > b.

2. PROOFS OF THEOREMS

In order to prove Theorem 1.1, we make use of the following elementary lemma which can be found in [3, p. 395].

Lemma 2.1 ([3, p. 395]). Let the second derivative ofφ(x)be continuous withx∈ (−∞,∞) andφ(0) = 0. Define

(2.1) g(x) =





 φ(x)

x , x6= 0;

φ⁰(0), x= 0.

Thenφ(x)is strictly convex (concave) if and only ifg(x)is strictly increasing (decreasing) with x∈(−∞,∞).

Remark 2.2. A general conclusion was given in [7, p. 18]: A functionφis convex on[a, b]if and only if ^φ(x)−φ(x_x−x ⁰⁾

0 is nondecreasing on[a, b]for every pointx0 ∈[a, b].

Proof of Theorem 1.1. It is obvious that f_α(s) =

Rb a x^sdx Rb

a(α−x)^sdx

!¹_s

=

b^s+1−a^s+1 (α−a)^s+1−(α−b)^s+1

¹_s

= L_s(a, b) L_s(α−a, α−b).

Define fors∈(−∞,∞),

(2.2) ϕ(s) =









 ln

b^s+1−a^s+1 (α−a)^s+1−(α−b)^s+1

, s6=−1;

ln

ln(b/a)

ln[(α−a)/(α−b)]

, s=−1.

Then

(2.3) lnf_α(s) =





 ϕ(s)

s , s 6= 0;

ϕ⁰(0), s = 0.

In order to prove that lnf_α is strictly increasing (decreasing), it suffices to show that ϕ is strictly convex (concave) on(−∞,∞). A simple calculation reveals that

(2.4) ϕ(−1−s) =ϕ(−1 +s) +sln(α−a)(α−b)

ab ,

which implies thatϕ⁰⁰(−1−s) = ϕ⁰⁰(−1 +s), and ϕ has the same convexity (concavity) on both(−∞,−1)and(−1,∞). Hence, it is sufficient to prove thatϕis strictly convex (concave) on(−1,∞).

A computation yields

ϕ⁰(s) = b^s+1lnb−a^s+1lna

b^s+1−a^s+1 −(α−b)^s+1ln(α−b)−(α−a)^s+1ln(α−a) (α−b)^s+1−(α−a)^s+1 , (s+ 1)²ϕ⁰⁰(s) = (s+ 1)²

"

−a^s+1b^s+1(ln^a_b)²

(b^s+1−a^s+1)² +(α−a)^s+1(α−b)^s+1(ln_α−a^α−b)² [(α−a)^s+1−(α−b)^s+1]²

#

=−(^a_b)^s+1[ln(^a_b)^s+1]²

[1−(^a_b)^s+1]² +(_α−a^α−b)^s+1[ln(_α−a^α−b)^s+1]² [1−(_α−a^α−b)^s+1]² . Define for0< t <1,

(2.5) ω(t) = t(lnt)²

(1−t)². Differentiation yields

(2.6) (1−t)tlntω⁰(t)

ω(t) = (1 +t) lnt+ 2(1−t) =−

∞

X

n=2

n−1

n(n+ 1)(1−t)ⁿ⁺¹ <0, which implies thatω⁰(t)>0for0< t <1. It is easy to see that

(2.7) 0<a b

s+1

<

α−b α−a

s+1

<1 for [a, b]⊆ 0,α

2 i

, s >−1,

(2.8) 0<

α−b α−a

s+1

<a b

s+1

<1 for [a, b]⊆hα 2, α

, s >−1,

and therefore ϕ⁰⁰(s) > 0 for [a, b] ⊆ (0,^α₂] and s > −1, ϕ⁰⁰(s) < 0 for [a, b] ⊆ [^α₂, α)and s > −1. Thenϕ is strictly convex (concave) on(−1,∞)for [a, b] ⊆ (0,^α₂] ([a, b] ⊆ [^α₂, α))

respectively. By Lemma 2.1 above, Theorem 1.1 holds.

Sincef_α(s)is a strictly increasing (decreasing) function for[a, b] ⊆ (0,^α₂] ([a, b] ⊆[^α₂, α)), puts =−2,−1,0,1respectively. The inequalities (1.6) are deduced.

Then, let(α−x) = tand apply it to the function ^Rb

ax^sdx Rb

a(α−x)^sdx

¹_s

. We get Corollary 1.3.

Proof of Theorem 1.4. Using an analogous method of proof to that of Theorem 1.1, we get

f(s) =

1 b−a

Rb ax^sdx

1 d−c

Rd c x^sdx

!¹_s

=

" _b^s+1_−a^s+1

(s+1)(b−a) d^s+1−c^s+1 (s+1)(d−c)

#¹s

=

(d−c) (b−a)

(b^s+1−a^s+1) (d^s+1−c^s+1)

¹_s

= L_s(a, b) L_s(c, d). LetM = ^(d−c)_(b−a), and define fors ∈(−∞,∞),

(2.9) ϕ(s) =









 ln

Mb^s+1−a^s+1 d^s+1−c^s+1

, s 6=−1;

ln

Mln(b/a) ln(d/c)

, s =−1.

Then

(2.10) lnf(s) =





 ϕ(s)

s , s6= 0;

ϕ⁰(0), s= 0,

andϕhas the same convexity (concavity) on both(−∞,−1)and(−1,∞).

A computation yields

(s+ 1)²ϕ⁰⁰(s) =−(^a_b)^s+1[ln(^a_b)^s+1]²

[1−(^a_b)^s+1]² + (_d^c)^s+1[ln(_d^c)^s+1]² [1−(_d^c)^s+1]² . Define for0< t <1,

(2.11) ω(t) = t(lnt)²

(1−t)².

Differentiation yieldsω⁰(t)>0for0< t <1. It is easy to see that

(2.12) 0<a

b s+1

<c d

s+1

<1 for ad < bc, s > −1,

(2.13) 0<c

d s+1

<a b

s+1

<1 for ad > bc, s > −1,

and thereforeϕ⁰⁰(s) > 0 forad < bcand s > −1, ϕ⁰⁰(s) < 0forad > bc ands > −1Then ϕ is strictly convex (concave) on (−1,∞) for ad < bc (ad > bc) respectively. The proof is

complete.

In Theorem 1.4, leta =c. Thenf(s)is a strictly increasing function ford < b, or a strictly decreasing function ford > b. Thus Corollary 1.5 holds.

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